Talk:Compound Poisson distribution

From Wikipedia, the free encyclopedia

WikiProject Mathematics
This article is within the scope of WikiProject Mathematics, which collaborates on articles related to mathematics.
Mathematics rating: Stub Class Low Priority  Field: Probability and statistics

I assume that E[Y] = λ * E[X]

What is Var[Y] in terms of the distribution of X? Say, if X has a gamma distribution.

[edit] Some properties

E[Y] = E[E[Y | N]] = λE[X]

Var[Y] = Var[E[Y | N]] + E[Var[Y | N]] = λ{E2[X] + Var[X]}

The cumulant generating function K_Y(t)=\mbox{ln} E[e^{tY}]=\mbox{ln} E[E[e^{tY}|N]]=\mbox{ln} E[e^{NK_X(t)}]=K_N(K_X(t))

One could add to the above, that if N has a Poisson distribution with expected value 1, then the moments of X are the cumulants of Y. Michael Hardy 20:39, 23 Apr 2005 (UTC)